Page 71 - Biosystems Engineering

52 Chapter Two

If we define the magnitude and phase of these vectors s and s as
p z
follows:

s = r and s = r
p i p i z i z i
(2.53)
s = ϕ and s = ϕ
p i p i z i z i
Equation (2.37) can be rewritten as
ss ... s rr ... r z ( j ϕ + ϕ + ... ϕ+ − ϕ − ϕ − ... −ϕϕ )
Gj )ω= z 1 z 2 z m = zz 2 m e z 1 z 2 z m p 1 p 2 p n (2.54)
(
1
ss ... s r rr ... r
1
p 1 p 2 p n pp 2 p n
From which the magnitude| (Gjω )|can be calculated as
rr ...r z
Gjω=
zz
|( )| 1 2 m (2.55)
rr ...r
1
pp 2 p n
ω
(
and the phase Gj ) is calculated as
ω
Gj ) (ϕ + ϕ + ... ϕ+ − ϕ − ϕ − ... ϕ− ) (2.56)
=
(
z 1 z 2 z m p 1 p 2 p n
Equations (2.55) and (2.56) demonstrate the advantage of using Bode
diagrams in this form. Because the magnitude is plotted on a loga-
rithmic scale, the overall magnitude and the phase information can
both be obtained from the component parts by a graphical addition of
the contributions by the different poles and zeros.
The zeros z , z ,…, z and the poles p , p ,…, p of the factorized
1 2 m 1 2 n
transfer function G(s) will each be real (including zero) or complex
with a complex conjugate, and thus G(s) can generally be considered
to be composed entirely of terms of the four types appearing in the
numerator or the denominator:
s + 2ζω s + ω 2
2
Ks;;1 + τ s; n n (2.57)
ω 2
n
Thus, G(jω) is a composition of multiples or quotients of terms of
the following form (Schwarzenbach and Gill 1978):
ω 2 − ω + 2 j ζ ωω
2
Kj ;1 + jωτ ; n n (2.58)
ω
;
ω 2
n
Constant Term (Gain Term)
Gs() = K G j ) = K (2.59)
ω
(
;